Atypical definition of decreasing function

[rahl__]

1 few days ago i saw a "strange" definition of a decreasing function in the web, but i can't find it now. there were three relationships, and when showing that one implies another, you could tell that the function is decreasing. one relationship looked like this:
$$f(x)=\frac{1}{x}$$
does it look familiar?

2 there is a function:
$$f(x)=\frac{2x+cos {x}}{3+x^{2}}$$
how can find if it is periodic[al?]? I've heard that the polynomial of an odd degree is not periodic[al], can I use this principle to define whether that function is periodic[al] or not? does this principle say[tell? sorry for this ungrammatical statement], that there are some polynomials of an even degree that are periodic[al]?

3 is it spelt periodic or periodical? ;/

 

[quasar987]

2 What you have here is not a polynomial. It's not even a quotient of polynomials. The general idea when trying to get determinethe period (if it exists) of a function is simply to write f(x) = f(x+L) and solve for L. If L is of the form kP (k integer), then P is your period.

 

[tacman]

The definition of a decreasing function may vary depending on the source, so it is possible that the definition you saw was not the traditional one. It is important to always refer to reliable sources when looking for mathematical definitions and concepts.

Regarding the given function f(x)=\frac{1}{x}, it is indeed a decreasing function as the value of the function decreases as x increases. This can be seen from the fact that as x increases, the denominator also increases, causing the overall value of the function to decrease.

For the second question, to determine if a function is periodic, you can check if it repeats itself after a certain interval. In the case of f(x)=\frac{2x+cos {x}}{3+x^{2}}, it is not a periodic function as it does not repeat itself after a specific interval. The principle you mentioned regarding polynomials of odd degree not being periodic is correct. However, there are also polynomials of even degree that are not periodic, so it is not a definitive rule to determine periodicity.

Lastly, the correct spelling is "periodic." "Periodical" can also be used, but it is less commonly used in mathematics.